Wave activity for large-amplitude disturbances described by the primitive equations on the sphere

Pseudomomentum and pseudoenergy are both measures of wave activity for disturbances in a fluid, relative to a notional background state. Together they give information on the propagation, growth, and decay of disturbances. Wave activity conservation laws are most readily derived for the primitive equations on the sphere by using isentropic coordinates. However, the intersection of isentropic surfaces with the ground (and associated potential temperature anomalies) is a crucial aspect of baroclinic wave evolution. A new expression is derived for pseudoenergy that is valid for large-amplitude disturbances spanning isentropic layers that may intersect the ground. The pseudoenergy of small-amplitude disturbances is also obtained by linearizing about a zonally symmetric background state. The new expression generalizes previous pseudoenergy results for quasigeostrophic disturbances on the β plane and complements existing large-amplitude results for pseudomomentum. The pseudomomentum and pseudoenergy diagnostics are applied to an extended winter from the European Centre for Medium-Range Weather Forecasts Interim Re-Analysis data. The time series identify distinct phenomena such as a baroclinic wave life cycle where the wave activity in boundary potential temperature saturates nonlinearly almost two days before the peak in wave activity near the tropopause. The coherent zonal propagation speed of disturbances at tropopause level, including distinct eastward, westward, and stationary phases, is shown to be dictated by the ratio of total hemispheric pseudoenergy to pseudomomentum. Variations in the lower-boundary contribution to pseudoenergy dominate changes in propagation speed; phases of westward progression are associated with stronger boundary potential temperature perturbations.

A three-dimensional hydrodynamic model with generic vertical coordinate

A three-dimensional primitive equation model and its application to a tidal estuary is described. The model solves the primitive equations for incompressible fluids with Boussinesq and hydrostatic approximations. The discretization is based on the finite volume method and allows a general vertical coordinate. The computational code is implemented in such a way that different vertical coordinates can be used in different parts of the domain. The model was designed to be able to simulate the flow both in the open ocean and in coastal and estuarine zones and can be coupled in a simple way to ecological models. The model was implemented successfully in several estuarine and coastal areas. Results are show for the Sado estuary in Portugal to illustrate model accuracy and potential. Quantitative validation is based on field data (water levels and velocities) while qualitative verification is based on the analysis of secondary flows.

Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi-hydrostatic and non-hydrostatic

We study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical geopotential and shallow atmosphere approximations in addition to the hydrostatic approximation. As is well known, a consistent application of the shallow atmosphere approximation requires omission of those Coriolis terms that vary as the cosine of latitude and of certain other terms in the components of the momentum equation. An approximate model is here regarded as consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity, and (following R. Müller) if its momentum component equations have Lagrange's form. Within these criteria, four consistent approximate global models, including the HPEs themselves, are identified in a height-coordinate framework. The four models, each of which includes the spherical geopotential approximation, correspond to whether the shallow atmosphere and hydrostatic (or quasi-hydrostatic) approximations are individually made or not made. Restrictions on representing the spatial variation of apparent gravity occur. Solution methods and the situation in a pressure-coordinate framework are discussed. © Crown copyright 2005.

Introduction and the problems of weather interpretation

Numerical forecasts of the atmosphere based on the fundamental dynamical and thermodynamical equations have now been carried for almost 30 years. The very first models which were used were drastic simplifications of the governing equations and permitting only the prediction of the geostrophic wind in the middle of the troposphere based on the conservation of absolute vorticity. Since then we have seen a remarkable development in models predicting the large-scale synoptic flow. Verification carried out at NMC Washington indicates an improvement of about 40% in 24h forecasts for the 500mb geopotential since the end of the 1950’s. The most advanced models of today use the equations of motion in their more original form (i.e. primitive equations) which are better suited to predicting the atmosphere at low latitudes as well as small scale systems. The model which we have developed at the Centre, for instance, will be able to predict weather systems from a scale of 500-1000 km and a vertical extension of a few hundred millibars up to global weather systems extending through the whole depth of the atmosphere. With a grid resolution of 1.5 and 15 vertical levels and covering the whole globe it is possible to describe rather accurately the thermodynamical processes associated with cyclone development. It is further possible to incorporate sub-grid-scale processes such as radiation, exchange of sensible heat, release of latent heat etc. in order to predict the development of new weather systems and the decay of old ones. Later in this introduction I will exemplify this by showing some results of forecasts by the Centre’s model.

Planetary-wave-induced transport in the stratosphere

In the stratosphere, chemical tracers are drawn systematically from the equator to the pole. This observed Brewer–Dobson circulation is driven by wave drag, which in the stratosphere arises mainly from the breaking and dissipation of planetary-scale Rossby waves. While the overall sense of the circulation follows from fundamental physical principles, a quantitative theoretical understanding of the connection between wave drag and Lagrangian transport is limited to linear, small-amplitude waves. However, planetary waves in the stratosphere generally grow to a large amplitude and break in a strongly nonlinear fashion. This paper addresses the connection between stratospheric wave drag and Lagrangian transport in the presence of strong nonlinearity, using a mechanistic three-dimensional primitive equations model together with offline particle advection. Attention is deliberately focused on a weak forcing regime, such that sudden warmings do not occur and a quasi-steady state is reached, in order to examine this question in the cleanest possible context. Wave drag is directly linked to the transformed Eulerian mean (TEM) circulation, which is often used as a surrogate for mean Lagrangian motion. The results show that the correspondence between the TEM and mean Lagrangian velocities is quantitatively excellent in regions of linear, nonbreaking waves (i.e., outside the surf zone), where streamlines are not closed. Within the surf zone, where streamlines are closed and meridional particle displacements are large, the agreement between the vertical components of the two velocity fields is still remarkably good, especially wherever particle paths are coherent so that diabatic dispersion is minimized. However, in this region the meridional mean Lagrangian velocity bears little relation to the meridional TEM velocity, and reflects more the kinematics of mixing within and across the edges of the surf zone. The results from the mechanistic model are compared with those from the Canadian Middle Atmosphere Model to test the robustness of the conclusions.

The middle-atmosphere Hadley circulation and equatorial inertial adjustment

In the tropical middle atmosphere the climatological radiative equilibrium temperature is inconsistent with gradient-wind balance and the available angular momentum, especially during solstice seasons. Adjustment toward a balanced state results in a type of Hadley circulation that lies outside the “downward control” view of zonally averaged dynamics. This middle-atmosphere Hadley circulation is reexamined here using a zonally symmetric balance model driven through an annual cycle. It is found that the inclusion of a realistic radiation scheme leads to a concentration of the circulation near the stratopause and to its closing off in the mesosphere, with no need for relaxational damping or a rigid lid. The evolving zonal flow is inertially unstable, leading to a rapid process of inertial adjustment, which becomes significant in the mesosphere. This short-circuits the slower process of angular momentum homogenization by the Hadley circulation itself, thereby weakening the latter. The effect of the meridional circulation associated with extratropical wave drag on the Hadley circulation is considered. It is shown that the two circulations are independent for linear (quasigeostrophic) zonal-mean dynamics, and interact primarily through the advection of temperature and angular momentum. There appears to be no significant coupling in the deep Tropics via temperature advection since the wave-driven circulation is unable to alter meridional temperature gradients in this region. However, the wave-driven circulation can affect the Hadley circulation by advecting angular momentum out of the Tropics. The validity of the zonally symmetric balance model with parameterized inertial adjustment is tested by comparison with a three-dimensional primitive equations model. Fields from a middle-atmosphere GCM are also examined for evidence of these processes. While many aspects of the GCM circulation are indicative of the middle-atmosphere Hadley circulation, particularly in the upper stratosphere, it appears that the circulation is obscured in the mesosphere and lower stratosphere by other processes.

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

Intercomparison of general circulation models for hot extrasolar planets

We compare five general circulation models (GCMs) which have been recently used to study hot extrasolar planet atmospheres (BOB, CAM, IGCM, MITgcm, and PEQMOD), under three test cases useful for assessing model convergence and accuracy. Such a broad, detailed intercomparison has not been performed thus far for extrasolar planets study. The models considered all solve the traditional primitive equations, but employ di↵erent numerical algorithms or grids (e.g., pseudospectral and finite volume, with the latter separately in longitude-latitude and ‘cubed-sphere’ grids). The test cases are chosen to cleanly address specific aspects of the behaviors typically reported in hot extrasolar planet simulations: 1) steady-state, 2) nonlinearly evolving baroclinic wave, and 3) response to fast timescale thermal relaxation. When initialized with a steady jet, all models maintain the steadiness, as they should—except MITgcm in cubed-sphere grid. A very good agreement is obtained for a baroclinic wave evolving from an initial instability in pseudospectral models (only). However, exact numerical convergence is still not achieved across the pseudospectral models: amplitudes and phases are observably di↵erent. When subject to a typical ‘hot-Jupiter’-like forcing, all five models show quantitatively di↵erent behavior—although qualitatively similar, time-variable, quadrupole-dominated flows are produced. Hence, as have been advocated in several past studies, specific quantitative predictions (such as the location of large vortices and hot regions) by GCMs should be viewed with caution. Overall, in the tests considered here, pseudospectral models in pressure coordinate (PEBOB and PEQMOD) perform the best and MITgcm in cubed-sphere grid performs the worst.

Baroclinic instability on hot extrasolar planets

We investigate baroclinic instability in flow conditions relevant to hot extrasolar planets. The instability is important for transporting and mixing heat, as well as for influencing large-scale variability on the planets. Both linear normal mode analysis and non-linear initial value cal- culations are carried out – focusing on the freely-evolving, adiabatic situation. Using a high- resolution general circulation model (GCM) which solves the traditional primitive equations, we show that large-scale jets similar to those observed in current GCM simulations of hot ex- trasolar giant planets are likely to be baroclinically unstable on a timescale of few to few tens of planetary rotations, generating cyclones and anticyclones that drive weather systems. The growth rate and scale of the most unstable mode obtained in the linear analysis are in qual- itative, good agreement with the full non-linear calculations. In general, unstable jets evolve differently depending on their signs (eastward or westward), due to the change in sign of the jet curvature. For jets located at or near the equator, instability is strong at the flanks – but not at the core. Crucially, the instability is either poorly or not at all captured in simulations with low resolution and/or high artificial viscosity. Hence, the instability has not been observed or emphasized in past circulation studies of hot extrasolar planets.

The slowly evolving background state of the atmosphere

The theory of wave–mean flow interaction requires a partition of the atmospheric flow into a notional background state and perturbations to it. Here, a background state, known as the Modified Lagrangian Mean (MLM), is defined as the zonally symmetric state obtained by requiring that every potential vorticity (PV) contour lying within an isentropic layer encloses the same mass and circulation as in the full flow. For adiabatic and frictionless flow, these two integral properties are time-invariant and the MLM state is a steady solution of the primitive equations. The time dependence in the adiabatic flow is put into the perturbations, which can be described by a wave-activity conservation law that is exact even at large amplitude. Furthermore, the effects of non-conservative processes on wave activity can be calculated from the conservation law. A new method to calculate the MLM state is introduced, where the position of the lower boundary is obtained as part of the solution. The results are illustrated using Northern Hemisphere ERA-Interim data. The MLM state evolves slowly, implying that the net non-conservative effects are weak. Although ‘adiabatic eddy fluxes’ cannot affect the MLM state, the effects of Rossby-wave breaking, PV filamentation and subsequent dissipation result in sharpening of the polar vortex edge and meridional shifts in the MLM zonal flow, both at tropopause level and on the winter stratospheric vortex. The rate of downward migration of wave activity during stratospheric sudden warmings is shown to be given by the vertical scale associated with polar vortex tilt divided by the time-scale for wave dissipation estimated from the wave-activity conservation law. Aspects of troposphere–stratosphere interaction are discussed. The new framework is suitable to examine the climate and its interactions with disturbances, such as midlatitude storm tracks, and makes a clean partition between adiabatic and non-conservative processes.

Sensitivity and variability redux in hot-Jupiter flow simulations

We revisit the issue of sensitivity to initial flow and intrinsic variability in hot-Jupiter atmospheric flow simulations, originally investigated by Cho et al. (2008) and Thrastarson & Cho (2010). The flow in the lower region (~1 to 20 MPa) `dragged' to immobility and uniform temperature on a very short timescale, as in Liu & Showman (2013), leads to effectively a complete cessation of variability as well as sensitivity in three-dimensional (3D) simulations with traditional primitive equations. Such momentum (Rayleigh) and thermal (Newtonian) drags are, however, ad hoc for 3D giant planet simulations. For 3D hot-Jupiter simulations, which typically already employ strong Newtonian drag in the upper region, sensitivity is not quenched if only the Newtonian drag is applied in the lower region, without the strong Rayleigh drag: in general, both sensitivity and variability persist if the two drags are not applied concurrently in the lower region. However, even when the drags are applied concurrently, vertically-propagating planetary waves give rise to significant variability in the ~0.05 to 0.5 MPa region, if the vertical resolution of the lower region is increased (e.g. here with 1000 layers for the entire domain). New observations on the effects of the physical setup and model convergence in ‘deep’ atmosphere simulations are also presented.

Interaction of equatorial waves through resonance with the diurnal cycle of tropical heating

In this work, a new theoretical mechanism is presented in which equatorial Rossby and inertio-gravity wave modes may interact with each other through resonance with the diurnal cycle of tropical deep convection. We have adopted the two-layer incompressible equatorial primitive equations forced by a parametric heating that roughly represents deep convection activity in the tropical atmosphere. The heat source was parametrized in the simplest way according to the hypothesis that it is proportional to the lower-troposphere moisture convergence, with the background moisture state function mimicking the structure of the ITCZ. In this context, we have investigated the possibility of resonant interaction between equatorially trapped Rossby and inertio-gravity modes through the diurnal cycle of the background moisture state function. The reduced dynamics of a single resonant duo shows that when this diurnal variation is considered, a Rossby wave mode can undergo significant amplitude modulations when interacting with an inertio-gravity wave mode, which is not possible in the context of the resonant triad non-linear interaction. Therefore, the results suggest that the diurnal variation of the ITCZ can be a possible dynamical mechanism that leads the Rossby waves to be significantly affected by high frequency modes.

Resonant Wave Interactions in the Presence of a Diurnally Varying Heat Source

Resonant interactions among equatorial waves in the presence of a diurnally varying heat source are studied in the context of the diabatic version of the equatorial beta-plane primitive equations for a motionless, hydrostatic, horizontally homogeneous and stably stratified background atmosphere. The heat source is assumed to be periodic in time and of small amplitude [i.e., O(epsilon)] and is prescribed to roughly represent the typical heating associated with deep convection in the tropical atmosphere. In this context, using the asymptotic method of multiple time scales, the free linear Rossby, Kelvin, mixed Rossby-gravity, and inertio-gravity waves, as well as their vertical structures, are obtained as leading-order solutions. These waves are shown to interact resonantly in a triad configuration at the O(e) approximation, and the dynamics of these interactions have been studied in the presence of the forcing. It is shown that for the planetary-scale wave resonant triads composed of two first baroclinic equatorially trapped waves and one barotropic Rossby mode, the spectrum of the thermal forcing is such that only one of the triad components is resonant with the heat source. As a result, to illustrate the role of the diurnal forcing in these interactions in a simplified fashion, two kinds of triads have been analyzed. The first one refers to triads composed of a k = 0 first baroclinic geostrophic mode, which is resonant with the stationary component of the diurnal heat source, and two dispersive modes, namely, a mixed Rossby-gravity wave and a barotropic Rossby mode. The other class corresponds to triads composed of two first baroclinic inertio-gravity waves in which the highest-frequency wave resonates with a transient harmonic of the forcing. The integration of the asymptotic reduced equations for these selected resonant triads shows that the stationary component of the diurnal heat source acts as an ""accelerator"" for the energy exchanges between the two dispersive waves through the excitation of the catalyst geostrophic mode. On the other hand, since in the second class of triads the mode that resonates with the forcing is the most energetically active member because of the energy constraints imposed by the triad dynamics, the results show that the convective forcing in this case is responsible for a longer time scale modulation in the resonant interactions, generating a period doubling in the energy exchanges. The results suggest that the diurnal variation of tropical convection might play an important role in generating low-frequency fluctuations in the atmospheric circulation through resonant nonlinear interactions.

Resonant Wave Interactions in the Equatorial Waveguide

Weakly nonlinear interactions among equatorial waves have been explored in this paper using the adiabatic version of the equatorial beta-plane primitive equations in isobaric coordinates. Assuming rigid lid vertical boundary conditions, the conditions imposed at the surface and at the top of the troposphere were expanded in a Taylor series around two isobaric surfaces in an approach similar to that used in the theory of surface-gravity waves in deep water and capillary-gravity waves. By adopting the asymptotic method of multiple time scales, the equatorial Rossby, mixed Rossby-gravity, inertio-gravity, and Kelvin waves, as well as their vertical structures, were obtained as leading-order solutions. These waves were shown to interact resonantly in a triad configuration at the O(epsilon) approximation. The resonant triads whose wave components satisfy a resonance condition for their vertical structures were found to have the most significant interactions, although this condition is not excluding, unlike the resonant conditions for the zonal wavenumbers and meridional modes. Thus, the analysis has focused on such resonant triads. In general, it was found that for these resonant triads satisfying the resonance condition in the vertical direction, the wave with the highest absolute frequency always acts as an energy source (or sink) for the remaining triad components, as usually occurs in several other physical problems in fluid dynamics. In addition, the zonally symmetric geostrophic modes act as catalyst modes for the energy exchanges between two dispersive waves in a resonant triad. The integration of the reduced asymptotic equations for a single resonant triad shows that, for the initial mode amplitudes characterizing realistic magnitudes of atmospheric flow perturbations, the modes in general exchange energy on low-frequency (intraseasonal and/or even longer) time scales, with the interaction period being dependent upon the initial mode amplitudes. Potential future applications of the present theory to the real atmosphere with the inclusion of diabatic forcing, dissipation, and a more realistic background state are also discussed.

Interaction of equatorial waves through resonance with the diurnal cycle of tropical heating

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